Note on a conjecture of Bateman and Diamond concerning the abstract PNT with Malliavin-type remainder

TL;DR

This paper constructs a generalized number system with specific error bounds in prime and integer counting functions, providing motivation for a longstanding conjecture on the prime number theorem with Malliavin-type remainders.

Contribution

It generalizes a construction of Diamond, Montgomery, and Vorhauer to produce Beurling systems with precise asymptotic behaviors, supporting the Bateman-Diamond conjecture from 1969.

Findings

Existence of Beurling systems with controlled error terms in counting functions.

Demonstrates the conjecture's plausibility through explicit examples.

Connects generalized number systems to classical prime number theorem conjectures.

Abstract

Given $\beta\in(0,1)$, we show the existence of a Beurling generalized number system whose integer counting satisfies $N(x) = ax + O\bigl(x\exp(-c\log^{\beta} x)\bigr)$ for some $a>0$ and $c>0$, and whose prime counting function satisfies $\pi(x) = \mathrm{Li}(x) + \Omega\bigl(x\exp(-c'(\log x)^{\frac{\beta}{\beta+1}})\bigr)$ for some $c'>0$. This is done by generalizing a construction of Diamond, Montgomery, and Vorhauer. This Beurling system serves as additional motivation for a conjecture of Bateman and Diamond from 1969, concerning the PNT with Malliavin-type remainder.